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F-algebra

Collection

context $F$ in ${\bf C}\longrightarrow{\bf C}$
definiendum $\langle A,\alpha\rangle$ in $\text{it}$
postulate $\alpha:{\bf C}[FA,A]$

Discussion

Think types $\mathrm{a}$ and $\alpha$'s of type

type Algebra f a = f a -> a

Example

The following examples assume that ${\bf C}$ contains all the relevant ingredients (e.g. products).

  • Addition of natural numbers is a binary relation: $+:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$. Hence $\langle \mathbb{N},+\rangle$ is an $F$-algebra for the endofunctor with object map $FX:=X\times X$.
  • Group actions on $X$ are maps $m:G\times X\to X$, so consider $F_GX:=G\times X$. The first example is also an $F_\mathbb{N}$-algebra.

Reference

Wikipedia: F-algbera

Parents

Context

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